notes Succinct series

Thomas' Calculus: Early Transcendentals

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
SERIES We say that the series X n =0 a n converges to L , or X n =0 a n = L , in case the limit of the sequence of partial sums N X n =0 a n is L ; i.e., lim N →∞ N X n =0 = L . The series X n =0 a n converges in case to converges to some number L ; otherwise, it diverges . From the definition, we find that the geometric series X n =0 r n = 1 1 - r if | r | < 1 and diverges if | r | ≥ 1. It follows that X n = k r n = r k 1 - r if | r | < 1 by factoring out the common factor r k . Tests for series with positive terms comparison If 0 a n b n and the big series X n =0 b n converges, then the small series X n =0 a n converges. (But if the big series diverges, this gives no information about the small series.) limit comparison (very useful) If 0 < a n , 0 < b n , and a n b n as n → ∞ (that is, lim n →∞ a n b n = 1), then the two series both converge or both diverge. Read as “is asymptotic to”, or “behaves like”. A polynomial in
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/12/2008 for the course MATH 104 taught by Professor Nelson during the Fall '07 term at Princeton.

Page1 / 2

notes Succinct series - SERIES We say that the series n=0...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online